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The dissection of the four pieces shown at left into the capital letter "T" shown at right.

For any two points x,y in X, there is an open set U such that x in U and y not in U or y in U and x not in U. A space fulfilling this axiom is called a T0-space.

A topological space fulfilling the T0-separation axiom: For any two points x,y in X, there is an open set U such that x in U and y not in U or y in U and x not in U. ...

For any two points x,y in X there exists two open sets U and V such that x in U and y not in U, and y in V and x not in V. A space satisfying this axiom is known as a ...

A T_1-space is a topological space fulfilling the T1-separation axiom: For any two points x,y in X there exists two open sets U and V such that x in U and y not in U, and y ...

Given any two distinct points x,y, there exist neighborhoods u and v of x and y, respectively, with u intersection v=emptyset. It then follows that finite subsets are closed.

A topological space fulfilling the T_2-axiom: i.e., any two points have disjoint neighborhoods. In the terminology of Alexandroff and Hopf (1972), a T_2-space is called a ...

X fulfils the T1-separation axiom and is regular. A space satisfying the T_3-separation axiom is said to be a T3-space.

A topological space fulfilling the T3-separation axiom: X fulfils the T1-separation axiom and is regular. According to the terminology of Alexandroff and Hopf (1972), ...

A topological space X fulfils the T1-separation axiom and is normal. A space fulfilling the T_4-separation axiom is said to be a T4-space.